Graph tan(theta)<0

Problem

tan(θ)<0

Solution

1. Identify the definition of the tangent function in terms of the unit circle coordinates (x,y) where tan(θ)=y/x

2. Determine the conditions for the inequality tan(θ)<0 to be true. This occurs when x and y have opposite signs.

3. Locate the quadrants where the coordinates have opposite signs. This happens in Quadrant II (where x<0 and y>0 and Quadrant IV (where x>0 and y<0.

4. Express the solution in terms of the angle θ In the first rotation [0,2*π) the tangent is negative in the intervals (π/2,π) and ((3*π)/2,2*π)

5. Generalize the solution for all real numbers by adding multiples of the period π

Final Answer

θ∈(⋃_k∈ℤ^)(k*π+π/2,k*π+π)

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